Solving two right triangles for x and y with tangent

Key concept: Work through each right triangle separately by matching the known side with the correct tangent relationship.

Step by step

Step 1: Identify the first triangle

The diagram gives a right triangle with a $35^\circ$ angle and a side of $10\text{ m}$. To find $x$, first decide which side the 10 m represents relative to the $35^\circ$ angle. In this type of question, the usual approach is to use the trig ratio that connects the known side to the unknown side.

If the 10 m side is adjacent to the $35^\circ$ angle and $x$ is opposite, then

$\tan 35^\circ = \frac{x}{10}$

so

$x = 10\tan 35^\circ \approx 7.0\text{ m}$

Step 2: Use the second triangle

For the second triangle, the $48^\circ$ angle is paired with the side length found from the first part, so the next step is to set up the correct trig ratio for $y$.

If $x$ is adjacent to the $48^\circ$ angle and $y$ is the hypotenuse, then use cosine:

$\cos 48^\circ = \frac{x}{y}$

so

$y = \frac{x}{\cos 48^\circ}$

Substitute the value of $x$:

$y \approx \frac{7.0}{\cos 48^\circ} \approx 10.5\text{ m}$

Step 3: Check the reasonableness

The first answer should be shorter than 10 m if it is an opposite side to a moderate acute angle. The second answer should be larger than $x$ if it is the hypotenuse. That matches the triangle relationships, so the results are consistent.

Important trig decisions

The main skill in this question is choosing the correct ratio before calculating. Do not start by typing numbers into the calculator randomly. First identify opposite, adjacent, and hypotenuse for each angle. Then select tangent, sine, or cosine based on which sides are involved.

Final answers

$\boxed{x\approx 7.0\text{ m}}$ $\boxed{y\approx 10.5\text{ m}}$

Pitfall alert

The most common problem here is mixing up which side is opposite, adjacent, or hypotenuse for each angle. Students often reuse the same trig ratio for both parts without re-checking the diagram, which can send the calculation in the wrong direction. Another issue is rounding too early: keep at least one extra decimal place until the end, then round to the nearest tenth of a metre. Also make sure the calculator is in degree mode, because the angles are given in degrees, not radians.

Try different conditions

If the 10 m side were the hypotenuse instead of an adjacent side, the setup would change completely. For example, you might use $\sin 35^\circ = x/10$ to find a leg, then use $\cos 48^\circ$ or $\tan 48^\circ$ depending on the new placement of the unknown. If the angles were changed to $30^\circ$ and $60^\circ$, the same triangle logic would apply, but the values would be different because each trig ratio depends on the specific angle. The method stays the same: label sides first, then choose the ratio.

Further reading

opposite side, adjacent side, trigonometric ratio